matrix algebra (linear algebra)

If we have the 2x6 matrix $\sigma$ (sorry do not know how to write this in neater way):

$\sigma$ = \begin{bmatrix}1&2&3&4&5&6 \\ 6&3&1&2&4&5\end{bmatrix}

and we were asked to find $\sigma^2$, $\sigma^3$ and ... and then find their signatures. I am okay with the signatures part but I have no idea what this $\sigma^2$ mean..

for $\sigma^2$ my professor got the matrix \begin{bmatrix}1&2&3&4&5&6 \\ 5&1&6&3&2&4\end{bmatrix}

• Squares are only defined for square matrices. This looks like a permutation problem. – Paul Feb 5 '15 at 19:53

It seems that this is not a $2\times 6$ matrix but a permutation. That is,

$$\sigma = \left(\begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & 6\\ 6 & 3 & 1 & 2 & 4 & 5\end{array}\right)$$

is notation for the function $\sigma : \{1, 2, 3, 4, 5, 6\} \to \{1, 2, 3, 4, 5, 6\}$ with

\begin{align*} \sigma(1) &= 6\\ \sigma(2) &= 3\\ \sigma(3) &= 1\\ \sigma(4) &= 2\\ \sigma(5) &= 4\\ \sigma(6) &= 5. \end{align*}

As such, the notation $\sigma^2$ means $\sigma\circ\sigma$. You can easily check that

\begin{align*} \sigma^2(1) &= 5\\ \sigma^2(2) &= 1\\ \sigma^2(3) &= 6\\ \sigma^2(4) &= 3\\ \sigma^2(5) &= 2\\ \sigma^2(6) &= 4 \end{align*}

so to denote this function $\sigma^2 : \{1, 2, 3, 4, 5, 6\} \to \{1, 2, 3, 4, 5, 6\}$ we would write

$$\sigma^2 = \left(\begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & 6\\ 5 & 1 & 6 & 3 & 2 & 4\end{array}\right).$$

• This is definitely a permutation problem. – ncmathsadist Feb 5 '15 at 19:58